Abstract

A finite-step iteration sequence for two finite families of asymptotically nonexpansive mappings is introduced and the weak and strong convergence theorems are proved in Banach space. The results presented in the paper generalize and unify some important known results of relevant scholars.

1. Introduction and Preliminaries

Throughout this work, we assume that is a real Banach space and is a nonempty subset of . A mapping is said to be asymptotically nonexpansive if there exists a sequence with such that

The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [1] in 1972 as an important generalization of the class of nonexpansive self-mappings, who proved that if is a nonempty closed convex subset of a real uniformly convex Banach space and is an asymptotically nonexpansive self-mapping of , then has a fixed point. Strong and weak convergence theorems for nonexpansive and asymptotically nonexpansive families of mappings and for single maps have been established by many authors (see [2–11]).

In [2], the authors introduced a multistep procedure defined by (2); under some conditions, they proved that the convergence of Mann-Ishikawa iterations is equivalent to the convergence of the multistep iteration in Banach spaces:
where the sequences , satisfy certain conditions.

In [3], Chidume and Ali studied a scheme defined by
where is a sequence in , . In a real uniformly convex Banach space , they proved the following: (i) a weak convergence theorem for finite families of asymptotically nonexpansive mappings where the dual space of satisfies the Kadec-Klee property; (ii) a strong convergence theorem if one member of the family of asymptotically nonexpansive maps satisfies a condition weaker than semicompactness.

Now, a finite-step iteration sequence for two finite families of asymptotically nonexpansive mappings is introduced as follows.

Let be a nonempty closed convex subset of a Banach space , and let , be two finite families of asymptotically quasi-nonexpansive mappings; the iterative sequence is defined by the iterative scheme
where with , is a nonnegative integer sequence in , and are fixed numbers.

Remark 1. In (4), taking , , , and for all , then we obtain (2); taking , , , and for all , then we obtain (3).

In this paper, the finite families of asymptotically quasi-nonexpansive mappings are defined in Banach spaces. Under certain conditions, we construct an iterative scheme and prove the following: (i) a weak convergence theorem for finite families of asymptotically quasi-nonexpansive mappings, where the uniformly convex Banach space satisfies Opial’s condition; (ii) necessary and sufficient conditions for convergence in real Banach spaces and a strong convergence theorem if the finite families of asymptotically quasi-nonexpansive mappings satisfy condition (). Our results generalize and unify many important known results of relevant scholars.

In order to prove the main results of this work, we need some basic concepts indicated as follows.

Let be a Banach space, and let be a nonempty closed convex subset of a . A mapping with domain and range in is said to be demiclosed at [3] if whenever is a sequence in such that and , then .

is said to satisfy Opial’s condition [5] if, for any sequence , implies that for all with , where denotes that converges weakly to .

Let be the self-mappings of and denotes the set of fixed points of .

Definition 2. is said to be a finite family of asymptotically nonexpansive mappings if there exists a sequence , , such that for all and .

Definition 3. is called a finite family of asymptotically quasi-nonexpansive mappings if there exists a sequence , , such that for all , and , , where .

Remark 4. The class of asymptotically quasi-nonexpansive mappings is a generalization of the class of nonexpansive mappings and asymptotically nonexpansive mappings.

Definition 5. are said to satisfy condition if there exists a nondecreasing function with , for all , such that for all , where and .

Lemma 6 (see [4]). Let and be two nonnegative real sequences satisfying
where ; then exists.

Lemma 7 (see [5]). Let , be two fixed numbers and let be a Banach space. Then is uniformly convex if and only if there exists a continuous, strictly increasing, and convex function with such that for all , and , where .

Lemma 8 (see [6]). Let be a nonempty closed subset of a uniformly convex Banach space , and let be an asymptotically nonexpansive mapping. Then is demiclosed at zero; that is, for each sequence , if converges weakly to and converges strongly to , then .

Lemma 9 (see [7]). Let be a Banach space which satisfies Opial’s condition and let be a sequence in . Let be such that and exist. If and are subsequences of which converge weakly to and , respectively, then .

Lemma 10. Let be a nonempty closed convex subset of a Banach space , and let be two finite families of asymptotically quasi-nonexpansive self-mappings of with sequences , and . Let be the sequence defined by (4), if the following conditions are satisfied:(i), ;(ii) for all and .Then and .

Proof. Let , for each . Since , for each , .Step 1. We prove that, for all and , and are existent and equal.It follows from (4) that we obtain that for any
and for , we have
Then, from (4), (6) and (7), we get
where . Since , from Lemma 6, for all , exists. Moreover, it is easy to see that also exist for all and .Step 2. We prove that and .Since is uniformly convex Banach space, from Lemma 7, letting , we get
for any and
From (9), we have
Since and are existent and equal, we have
Because is strictly increasing and continuous and ,
Further, similar to the computations above, using (10) and (11), we also can get for any
Hence, for all , we can obtain
Since for all , we have , and then
From (16) and (17), for
so
Hence from (16) and (18), for , we have
It follows from (17) and (20) that, for ,
Together with (17), for From (19) and (22), for any , we have
Together with (16) and (20), for , we have
Together with (16), for , we have
Since , together with (23) and (25), for , then

Theorem 11. Under the assumptions of Lemma 10, if is a uniformly convex Banach space satisfying Opial’s condition, then the sequence given by (4) converges weakly to a common fixed point of and .

Proof . By using the same proof as in Lemma 10, it can be shown that for any
So and are demiclosed at 0.Since is uniformly convex and is bounded, we may assume that as , without loss of generality. By Lemma 8, we have . Suppose that subsequences and of converge weakly to and , respectively. From Lemma 8, . By Lemma 10, and exist. It follows from Lemma 9 that . Therefore converges weakly to a common fixed point of and .

In this section, we prove strong convergence theorems of the iterative schemes (4) in Banach spaces.

Theorem 12. Under the assumptions of Lemma 10, the sequence given by (4) converges strongly to if and only if , where .

Proof . Necessity is obvious. We only prove the sufficiency. Suppose that . As proved in Lemma 10, for each , we have ; that is, . From Lemma 6, exists, based on the assumption that .Next, we can prove that is a Cauchy sequence in . In fact, for any ,
where . Hence for any positive integers , we have
Since , then . Thus, we get
where . So we have
This shows that is a Cauchy sequence in , since is a nonempty closed convex subset of a Banach space ; that is, is a complete space. Without loss of generality, we can assume that converges strongly to a common fixed point .

Theorem 13. Under the assumptions of Lemma 10, if satisfy condition (), then the sequence defined by (4) converges strongly to a common fixed point .

Proof. It follows from Lemma 10 that, for any , we have
Since satisfy condition (), we have .Since is a nondecreasing function with , for all , such that, for all , ; by Theorem 12, we obtain that converges strongly to a common fixed point .

Corollary 14. Under the assumptions of Lemma 10, the iteration sequence is defined as follows:
where with .(i)If satisfies Opial’s condition, then converges weakly to a common fixed point .(ii)If satisfy condition , then converges strongly to a common fixed point of .

Proof. By taking for all in (4), from Theorems 11 and 13, the conclusion of the corollary follows.

Corollary 15. Under the assumptions of Lemma 10, the iteration sequence is defined as follows:
where with .(i)If satisfies Opial’s condition, then converges weakly to a common fixed point .(ii)If satisfy condition , then converges strongly to a common fixed point of .

Proof. By taking , for all in (4), from Theorems 11 and 13, the conclusion of the corollary follows. This completes the proof.

Corollary 16. Let be a nonempty closed convex subset of a uniformly convex Banach space , and let be a family of asymptotically quasi-nonexpansive self-mappings of with sequences such that and . The iteration sequence is defined by (2) satisfying with .(i)If satisfies Opial’s condition, then converges weakly to a common fixed point of .(ii)If satisfies condition , then converges strongly to a common fixed point of .

Proof. By taking , , and for all in (4), we get (3). From Theorems 11 and 13, the conclusion of the corollary follows.

Corollary 17. Let be a nonempty closed convex subset of a uniformly convex Banach space , and let be a family of nonexpansive self-mappings of with sequences such that and . The iteration sequence is defined by (2) satisfying with .(i)If satisfies Opial’s condition, then converges weakly to a common fixed point of .(ii)If satisfies condition , then converges strongly to a common fixed point of .

Proof. By taking , , , and for all in (4), we get (2), which was introduced by Rhoades and Soltuz in [2]. From Theorems 11 and 13, the conclusion of the corollary follows.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (11226228), the Science and Technology Program Project of Henan Province (122300410256), and the Natural Science Foundation of the Education Department of Henan Province (2011B110025).